PHYSICAL REVIEW E 105, 044108 (2022) Editors’ Suggestion Exactly solvable percolation problems Fabian Coupette * and Tanja Schilling Institute of Physics, University of Freiburg, Hermann-Herder-Straße 3, 79104 Freiburg, Germany (Received 27 November 2021; accepted 31 January 2022; published 6 April 2022) We propose a simple percolation criterion for arbitrary percolation problems. The basic idea is to decompose the system of interest into a hierarchy of neighborhoods, such that the percolation problem can be expressed as a branching process. The criterion provides the exact percolation thresholds for a large number of exactly solved percolation problems, including random graphs, small-world networks, bond percolation on two-dimensional lattices with a triangular hypergraph, and site percolation on two-dimensional lattices with a generalized triangular hypergraph, as well as specific continuum percolation problems. The fact that the range of applicability of the criterion is so large bears the remarkable implication that all the listed problems are effectively treelike. With this in mind, we transfer the exact solutions known from duality to random lattices and site-bond percolation problems and introduce a method to generate simple planar lattices with a prescribed percolation threshold. DOI: 10.1103/PhysRevE.105.044108 I. INTRODUCTION Since its introduction in the late 1950s [1], the critical phenomenon of percolation has been examined extensively by both physicists (see [2] for a review) and mathematicians (see [3] for a review). Its practical relevance spans from net- work theory describing, e.g., the spreading of diseases [4,5] or the design of infrastructure [6] to material science, where, e.g., flow through porous media [7] and the conductivity of filler networks dispersed in insulators [8,9] are analyzed. Through the years a rich variety of theoretical tools have been developed and refined to tackle percolation in those diverse contexts, and even the problem statement itself has been mod- ified and extended in several different ways [10,11]. In two dimensions conformal field theory [12,13] has enabled the cal- culation of critical exponents, and lattice duality has been used to determine the critical point of the square lattice [3,14]. Due to the advance of computer power within the last half century, numerical methods and algorithmic optimization have gained increasing attention [15–18]. In the first part of this article we introduce an alternative characterization of percolation. Then we recall exact solutions to various structurally different percolation problems and the methods originally used to solve them, and we discuss their common denominator: the treelike structure, which allows for the application of our method. Then we show that the method can be used to design networks of a prescribed percolation threshold, and we extend the spectrum of exactly solvable percolation problems. II. MEAN FIELD PERCOLATION The idea of the approach we present in the following is based on an alternative characterization of percolation thresh- old which we will first derive and then apply to the full * fabian.coupette@physik.uni-freiburg.de range of exactly solved percolation problems. For the sake of readability, we initially define the required quantities only for homogeneous bond percolation in a discrete system. However, the generalization to site percolation, site-bond percolation, and also continuum problems is straightforward and will be exercised as the article progresses. On a connected graph G( V, E ) with vertices V and edges E a bond percolation model is defined by assigning a probability p to each edge of being open. Two vertices are connected if there is an open path between them, i.e., a sequence of open edges linking one vertex to the other. (Note that in the literature the term occupied is often used in place of open, in particular, in the context of site percolation rather than bond percolation problems. The meaning of the terms is the same. Here, we follow the nomenclature of the textbook by Grimmett [3].) The percolation probability ( p) is defined as the probabil- ity that an arbitrarily assigned vertex, which we call the origin O, is part of an infinite connected component. The percolation threshold p c demarcates the edge probability for which the percolation probability ceases to be zero, ( p) > 0 for p > p c , (1) ( p) = 0 for p < p c , (2) formally defined by [3] p c := sup{ p : ( p) = 0}. (3) For all systems we treat here, we can define a metric that characterizes the distance between two sites or particles. For G( V, E ) specifically we can take the number of edges visited on the shortest path between two given vertices as metric. Given a vertex A, all the edges incident to A together with the vertices linked by those edges form a subgraph we label the 1-neighborhood N 1 (A) of A. (See the top left panel of Fig. 1 for an illustration. The 1-neighborhood of the circle with the 2470-0045/2022/105(4)/044108(10) 044108-1 ©2022 American Physical Society